Differential coefficient

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

Together with Eos. (4-6) and (4-7), this definition allows ehmination of all the partial differential coefficients from the preceding equation ... 

The obvious next step is to ehminate the partial-differential coefficients in favor of measurable quantities. 

Of the five magnitudes p, v, , s, u, any two may be chosen for the independent variables x and y, and for each pair we shall, by mean s of (11), obtain a relation between the differential coefficients of two other magnitudes with respect to the chosen variables. These are deduced as follows ... 

Since the entropy is the partial differential coefficient of with r constant, or with p constant, with respect to T, the magnitudes and thermodynamic potentials at constant volume and at constant pressure respectively. 

The differential coefficient is total, since E does not depend on the extent of the chemical change. 

The adsorption-induced charging of the barriers of such adsorbents results in the change in heights of inter-crystalline barriers and transforms the profile of their distribution function. As the model suggests, it is this change in the distribution function of the heights of barriers, that is responsible for adsorption-induced change of such an important characteristic of polycrystal as the differential coefficients of its volt-ampere characteristics or, which is more convenient for our studies,... 

Comparisons with equation (30) reveal that Tammann replaced Tait s compression term, (v° - vP)/vuP by the corresponding differential coefficient (3v/3P)P/v. In the low pressure range (below 500 bars) that concerns the majority of workers,... 

This gives (L//G )v/(pg/Pl) = 16, when dG2F/dpG = 0- As the second differential coefficient is negative at this point, G2F is a maximum. 

The differential coefficients d[R]/df and d P]/df are the mathematical expressions for the slope of the tangent drawn to a curve at the time of interest. Similarly, the unique instantaneous rate of a reaction is defined as in Eq. 2, but with differential coefficients ... 

For the simple case of a planar, infinitely extended planar surface, the potential cannot change in the y and z direction because of the symmetry and so the differential coefficients with respect to y and z must be zero. We are left with the Poisson-Boltzmann equation which contains only the coordinate normal to the plane x ... 

But, if 5x —> 0, then this ratio tends to a definite quantity. This definite quantity is called the differential coefficient of fix) with respect to x and it is denoted by... 

For example, to find the differential coefficient of xe with respect to x, we have... 

We know that the differential coefficient of f x) with respect to x is ton... 

If u =f(x, y) be a continuous function of two independent variables x and y, then the differential cofficient w.r.t. x (taking y as constant) is called the partial derivative or partial differential coefficient of u w.r.t. x and is represented by different symbols such as... 

If it exists then, it is called the partial derivative or partial differential coefficient of u w.r.t. jc. It is denoted by,... 

Problem 8. Find the differential coefficients of the following sinx x... 

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